📚 KS3 CAIE Philosophy: Formula & Theorem Quick Reference Guide | KS3 CAIE 哲学：公式定理速查手册
This quick reference guide collects the core reasoning patterns, principles, and conceptual ‘formulas’ used in KS3 CAIE Philosophy. While philosophy does not use equations in the same way as mathematics, many arguments follow strict logical forms that can be represented as formulas. Mastering these will sharpen your analysis, help you construct strong arguments, and enable you to spot fallacies in any philosophical discussion. Each entry below is presented with a clear English explanation followed by its Chinese counterpart, making this a truly bilingual revision tool.
这份速查手册汇集了 KS3 CAIE 哲学课程中使用的核心推理模式、原理和概念性“公式”。虽然哲学并不像数学那样使用方程，但许多论证都遵循严格的逻辑形式，可以用公式来呈现。掌握这些内容将提升你的分析能力，帮助你构建有力的论证，并让你在任何哲学讨论中都能识别谬误。以下每个条目都配有清晰的英文解释和对应的中文解释，是一份真正的双语复习工具。
1. What is a Philosophical Argument? | 什么是哲学论证？
A philosophical argument is a structured set of statements where one or more premises are given in support of a conclusion. The premises are meant to provide reasons or evidence that make the conclusion likely or certain. For example, ‘All humans are mortal. Socrates is a human. Therefore, Socrates is mortal.’ This is not a quarrel but a rational attempt to prove a point.
哲学论证是一组结构化的陈述，其中一个或多个前提被用来支持一个结论。前提旨在提供理由或证据，使得结论成为可能或必然。例如，“所有人都会死。苏格拉底是人。所以，苏格拉底会死。”这不是争吵，而是理性的证明尝试。
2. Deductive and Inductive Reasoning | 演绎推理与归纳推理
Deductive reasoning aims to provide conclusive proof. If the premises are true and the logical form is valid, the conclusion must be true. The classic form is: Premise 1 → Premise 2 → Conclusion. Inductive reasoning, by contrast, provides probable support. It moves from specific observations to general conclusions, such as ‘The sun has risen every day in recorded history, so the sun will rise tomorrow.’ Deductive arguments guarantee truth when valid and sound; inductive arguments only make the conclusion likely.
演绎推理旨在提供决定性的证明。如果前提为真且逻辑形式有效，结论必然为真。其经典形式是：前提1 → 前提2 → 结论。归纳推理则提供或然性的支持，从具体的观察推出一般性结论，例如“有史以来太阳每天都升起，所以太阳明天也会升起。”演绎论证在有效且可靠时保证结论为真；归纳论证只能使结论具有一定可能性。
3. Modus Ponens (Affirming the Antecedent) | 肯定前件式
P → Q, P ⊢ Q
This is one of the most fundamental valid argument forms. It states: If P is true then Q is true. P is true. Therefore, Q is true. For instance, ‘If it is raining, the ground is wet. It is raining. So, the ground is wet.’ The formula is foolproof: whenever the conditional premise and the antecedent are true, the consequent follows logically.
这是最基本的有効论证形式之一。它陈述：如果 P 为真，则 Q 为真。P 为真。因此，Q 为真。例如，“如果下雨，地面就湿。下雨了。所以，地面湿了。”这个公式万无一失：只要条件前提和前件为真，后件就在逻辑上必然得出。
4. Modus Tollens (Denying the Consequent) | 否定后件式
P → Q, ¬Q ⊢ ¬P
Another rock-solid deductive form. It says: If P then Q. Q is false. Therefore, P must be false. Example: ‘If the battery is charged, the phone will turn on. The phone does not turn on. So, the battery is not charged.’ Be careful not to confuse this with the fallacy of denying the antecedent.
另一个稳固的演绎形式。它表明：如果 P 则 Q。Q 为假。因此，P 必定为假。示例：“如果电池有电，手机就会开机。手机没有开机。所以，电池没有电。”注意不要把这个与否定前件的谬误相混淆。
5. Hypothetical Syllogism | 假言三段论
P → Q, Q → R ⊢ P → R
This formula chains conditionals together. If P implies Q, and Q implies R, then P implies R. For example, ‘If I study, I will pass the exam. If I pass the exam, I will graduate. Therefore, if I study, I will graduate.’ The chain must be linked without gaps for the argument to hold.
这个公式将条件句串联起来。如果 P 蕴含 Q，Q 蕴含 R，那么 P 蕴含 R。例如，“如果我学习，我就会通过考试。如果我通过考试，我就会毕业。因此，如果我学习，我就会毕业。”论证要成立，链条中间不能有断开。
6. Disjunctive Syllogism | 选言三段论
P ∨ Q, ¬P ⊢ Q
When you have an ‘either–or’ statement and one option is eliminated, the other must be true. The formula: P or Q. Not P. Therefore, Q. Example: ‘The cake is either chocolate or vanilla. It is not chocolate. So, it is vanilla.’ This only works if the disjunction genuinely covers all possibilities (an exclusive ‘or’).
当你有一个“要么……要么……”的陈述，并且其中一个选项被排除时，剩下的那个必然为真。公式：P 或 Q。非 P。因此，Q。示例：“蛋糕要么是巧克力味，要么是香草味。它不是巧克力味。所以，它是香草味。”这只有在选言支确实覆盖所有可能性（即排斥性的“或”）时才有效。
7. Constructive Dilemma | 二难推理
(P → Q) ∧ (R → S), P ∨ R ⊢ Q ∨ S
A more complex valid form often used in ethical debates. The pattern: If P then Q, and if R then S. Either P or R is true. Therefore, either Q or S is true. For instance, ‘If we increase taxes, public services improve. If we cut spending, the deficit falls. We must either increase taxes or cut spending. So, either public services improve or the deficit falls.’ The conclusion follows inevitably.
这是更复杂的有效形式，常用于伦理辩论。模式为：如果 P 则 Q，并且如果 R 则 S。要么 P 为真，要么 R 为真。因此，要么 Q 为真，要么 S 为真。例如，“如果增税，公共服务会改善。如果削减开支，赤字会下降。我们必须要么增税，要么削减开支。所以，要么公共服务改善，要么赤字下降。”结论必然得出。
8. Reductio ad Absurdum | 归谬法
Assume ¬P, derive contradiction, ∴ P
This is proof by contradiction. To prove a statement P, you temporarily assume its negation (not-P) and show that this leads to a logical contradiction or absurdity. Since contradictions cannot be true, the assumption must be false, so P is true. Example: To prove ‘There is no largest prime number,’ assume there is a largest prime, then derive a contradiction by constructing a larger one. This technique is widely used in philosophy of religion and mathematics.
这是反证法。要证明一个陈述 P，你暂时假定它的否定（非 P），并表明这会导致逻辑矛盾或荒谬。既然矛盾不可能为真，这个假定必定为假，因此 P 为真。示例：要证明“不存在最大的质数”，先假定有一个最大的质数，然后通过构造一个更大的质数推导出矛盾。这种方法在宗教哲学和数学中广泛使用。
9. Common Fallacies: Affirming the Consequent and Denying the Antecedent | 常见谬误：肯定后件与否定前件
Affirming the Consequent: The invalid form P → Q, Q ⊢ P. Just because Q is true does not mean P must be true. Example: ‘If it rained, the grass is wet. The grass is wet, therefore it rained.’ (The grass could be wet from sprinklers.)
肯定后件谬误：无效形式 P → Q，Q ⊢ P。仅仅是 Q 为真，并不意味着 P 必然为真。示例：“如果下雨，草地就湿。草地湿了，所以下过雨。”（草地可能是洒水器弄湿的。）
Denying the Antecedent: The invalid form P → Q, ¬P ⊢ ¬Q. Even when P is false, Q might still be true. Example: ‘If it is a dog, it is an animal. It is not a dog, therefore it is not an animal.’ (It could be a cat.)
否定前件谬误：无效形式 P → Q，¬P ⊢ ¬Q。即使 P 为假，Q 仍可能为真。示例：“如果它是狗，它就是动物。它不是狗，所以它不是动物。”（它可能是猫。）
10. Ockham’s Razor | 奥卡姆剃刀
Entities should not be multiplied beyond necessity.
This principle of parsimony states that, when faced with competing explanations, the one that makes the fewest assumptions is usually preferred. It does not claim the simpler theory is always true, but that it is more rational to adopt it until evidence demands complexity. In philosophy, this razor trims away unnecessary metaphysical entities. For example, if two theories explain the same data equally well, choose the one with fewer unobservable beings or forces.
这条简约原则指出，面对多种解释时，通常应该选择所作的假定最少的那一个。它并不是说更简单的理论总是真实的，而是说在证据迫使增加复杂性之前，采纳最简单的理论更合理。在哲学中，这把剃刀削去不必要的形而上学实体。例如，如果两个理论同样很好地解释相同的数据，就选择那个包含较少不可观察存在或力量的理论。
11. Cogito Ergo Sum (Descartes) | 我思故我在
I think, therefore I am. (Dubito, ergo cogito, ergo sum.)
Descartes’ foundational ‘formula’ emerged from his method of radical doubt. Even if an evil demon is deceiving him about everything, the very act of being deceived presupposes that he exists as a thinking thing. This is not a formal logical syllogism but an immediate self-evident intuition: the act of doubting confirms one’s own existence. In philosophical shorthand, it is the first certainty that resists all sceptical challenges.
笛卡尔的基础“公式”源于他彻底怀疑的方法。即使有一个邪恶的恶魔在一切事情上欺骗他，被欺骗这一行为本身就预设了他作为一个思想者的存在。这不是一个形式逻辑上的三段论，而是一种直接的、自明的直觉：怀疑的行为本身就证实了自身的存在。用哲学的简化表述来说，这是第一个能够抵御所有怀疑论挑战的确定性。
12. The Greatest Happiness Principle (Utilitarianism) | 最大幸福原则（功利主义）
Action is right if it produces the greatest happiness for the greatest number.
This is the core normative formula of classical utilitarianism, proposed by Jeremy Bentham and John Stuart Mill. The moral worth of an action is calculated by summing up the pleasure it produces and subtracting the pain it causes, counting all affected individuals equally. Although not a mathematical equation, Bentham’s ‘felicific calculus’ attempted to weigh intensity, duration, certainty, and extent of pleasure. The principle functions as a decision-making theorem in ethics.
这是古典功利主义的核心规范性公式，由边沁和密尔提出。一个行为的道德价值是通过加总它所产生的快乐并减去它造成的痛苦来计算的，所有受影响的个体都被同等计算。虽然不是一个数学等式，边沁的“快乐计算法”试图权衡快乐的强度、持续时间、确定性和范围。这个原则在伦理学中充当着一种决策定理。
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